II. To prove that the area of the circle is the common limit of S and S' when the number of sides is indefinitely increased.

(R)2. SUG. 4. = S' (R')2

S

Conclusion.

Auth. See Part I for method.

THEOREM IX

215. The circumferences of two circles have the same ratio as their radii.

Hyp. Let C and C" be the circumferences, R and R' the radii, of two circles O and O'.

SUG. 3. By theory of limits show that C=C' × and there

R

R'

216. Cor.

The areas of two circles have the same ratio as the

squares of their radii.

SUG. See Theo. VIII, Part II.

217. Scho. The ratio of the circumference of a circle to its diameter is a fixed quantity. This ratio is designated by π. and .'. C = πd = 2 πι. The numerical value of T

Hence

с

d

is approximately 3.14159 +, or about 34.

NOTE. See Problem XXV, page 108, for proof regarding value of π.

THEOREM X

218. The area of a circle equals half the product of the circumference and the radius.

SUG. Inscribe a regular polygon, and prove that P× a (apothem) approaches as its limit C × r.

219. Scho. If C = 2πr, then Cx (the area of circle) 1 (2 πr') × r = πr2.

=

EXERCISES

157. Similar arcs are to each other as their radii, and similar sectors as the squares of their radii.

158. If the radius of a circle is 4, find its circumference and area. 159. If the circumference of a circle is 30, find its radius and area. 160. If the diameter of a circle is 26, find the length of an arc of 90°.

220. Problem XXIV. Given the radius and side of a regular inscribed polygon, to compute the side of a regular inscribed polygon of double the number of sides.

Hyp. In 0, let AB be the side of a regular inscribed polygon and let the radius OX be 1 to AB, intersecting chord AB at Y and AB at X.

To find value of AX, the side of a regular inscribed polygon of double the number of sides.

Draw AO. In rt. ▲ AOY,

Οι

OY2= A02-AY'. Auth. But AYAB. Auth.

XY = OX-OY= r - †√4 r2 — AB.

.'. In rt. ^AXY, AX2=AY2+XY2=AB2 + (r− + √4 ro — AB3)2.

4

=

.. AX r (2 r — √4r 2

- AB3.

Q.E.F.

221. Cor. If r=1 and if the side of the inscribed polygon is represented by s, the side of the inscribed polygon of double the number of sides equals √2 −√4 — s2.

EXERCISES

161. Find the radius of a circle whose area is 6 sq. in.

162. What is the area of a circle circumscribed about a square whose side is 3?

222. Problem XXV. To compute, approximately, the value of π. Hyp. Let c and d represent, respectively, the circumference and diameter of a circle whose radius is unity.

Since the perimeter of a regular inscribed polygon, by indefinitely increasing the number of its sides, approaches the circumference of the circle as its limit, it follows that

perimeter of polygon

diameter of circle

approaches as its limit C

d'

as the number of sides of the polygon is indefinitely increased. Let S6, S12, etc., represent the side of a regular hexagon, of a regular dodecagon, etc., P6, P12, etc., their perimeters,

$48 =

√2-√4-(.26105238)=.13080626; =3.13935020,

and by continuing the process

P48

d

163. How many degrees are there in an arc 18 in. long of a circumference whose radius is 5 ft. ?

164. The ratio of the radii of two similar segments is 3:5. What is the ratio of their areas?

165. The circumference of a circle is 8. Find the circumference of a circle having twice the area of the given circle.

166. A circle has an area of 60 sq. in. What is the length of an arc of 40°? 167. The side of a square is 8. Find the circumferences of its inscribed and circumscribed circles.

168. A circular grass plot, 100 ft. in diameter, is surrounded by a walk 4 ft. wide. Find the area of the walk.

INDEX TO PROBLEMS OF CONSTRUCTION

Problem

I. To draw a perpendicular to a straight line
at some point in it

Problem II. To bisect a given angle

Problem III.

Problem IV.

Problem V.

Problem VI.

PAGE

6

6

To construct an angle equal to a given angle 10
To bisect a given straight line

Given two sides and the angle to be included
by them, to construct the triangle.
Given two angles and the side to be included
by them, to construct the triangle
Problem VII. Given three sides of a triangle, to construct
the triangle

Problem VIII. To draw a line parallel to a given line
through a given point

Problem IX. To draw a line perpendicular to a given line
from an outside point

Problem X. Given two angles of a triangle, to find the

third

Problem XI. To draw a tangent to a given circle
Problem XII. To find the centre of curvature of a given

Problem XIII. To draw a circle tangent to a given circle
Problem XIV. To divide a given line into any number of

equal parts

12

63

74